stat611hw1win09 - Statistics 611: Homework 1 (1) Keeners...

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Statistics 611: Homework 1 (1) Keener’s notes Chapter 9: 7, 11, 12, 15, 26. (2) Consider a family of random variables { X n } where X n has the following probability mass function: P ( X n = n γ ) = n - δ and P ( X n = 0) = 1 - n - δ , where δ > 0. For fixed δ , for what values of γ is the family (a) { X n } uniformly integrable, (b) { X n } not uniformly integrable? (3) Consider a family of estimates { T n } of a real parameter μ satisfying n ( T n - μ ) N (0 2 ). Let g be a twice continuously differentiable transformation with g 0 ( μ ) = 0. Show that for an appropriate sequence { α n } , which you need to identify, the sequence α n ( g ( T n ) - g ( μ )) has a limit distribution and identify this limit. If X 1 ,X 2 ,... are i.i.d. with mean π/ 2 and variance 1, deduce the limiting behavior of sin X n using this general result. (4) Consider the linear regression model: Y i = α + β X i + ± i , i = 1 , 2 ,... , where X 1 ,X 2 ,... is a sequence of constants and { ± i } is an i.i.d. sequence of mean 0 errors with common variance
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stat611hw1win09 - Statistics 611: Homework 1 (1) Keeners...

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